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It seems that you've got factorization of maps covered, so let me address the question of why canonical quotient maps and canonical inclusions are "better".

Given a set $X$, in general there is a proper class of injections $Y \to X$. However, many of these are isomorphic, where injections $i : Y \to X$ and $j : Z \to X$ are isomorphic if there is an isomorphism $k : Y \to Z$ such that $i = j \circ k$. The isomorphism classes of injections into $X$ are the subobjects of $X$. In fact, there are only set-many subobjects of $X$ (in category-theoretic language, sets form a well-powered category). It is pesky to work with set-many proper equivalence classes, so we instead look for a set $P(X)$ of injections into $X$, one from each isomorphism class. We may additionally require some nice properties, for instance, if $i : X \to Y$ is in $P(Y)$ and $j : Y \to Z$ is in $P(Y)$, we would expect $j \circ i : X \to Z$ to be in $P(Z)$. One can come up with a wish list of such nice closure conditions, here's another one: if $i : Y \to X$ and $j : Z \to X$ are in $P(X)$, and there is (a unique) $k : X \to Z$ such that $i = j \circ k$, then $k$ is in $P(Z)$.

We know the answer, of course, just take $P(X)$ to be the canonical subset inclusions into $X$. This is not the only choice of such representative inclusions, but it's a pretty good one.

We may therefore say that the canonical inclusions of subsets are "better" because they are the canonical representatives of subobjects (equivalence classes of injections).

The answer for quotient maps and surjections is dual. Consider equivalence classes of surjections, quotiented by isomorphism. There are only set-many such classes, therefore sets form a well-copowered category. (Some people say "cowell-powered" but then why not call it "ill-powered"?) This time we look for a set $Q(X)$ of surjections from $X$, each representing one equivalence class of surjections from $X$. We may take $Q(X)$ to be the set of all canonical quotient maps $X \to X/{\sim}$, or just the set of all equivalence classes on $X$. Once again, canonical quotient maps are "better" because they are the distinguished representatives of isomorphism classes of surjections.

It seems that you've got factorization of maps covered, so let me address the question of why canonical quotient maps and canonical inclusions are "better".

Given a set $X$, in general there is a proper class of injections $Y \to X$. However, many of these are isomorphic, where injections $i : Y \to X$ and $j : Z \to X$ are isomorphic if there is an isomorphism $k : Y \to Z$ such that $i = j \circ k$. The isomorphism classes of injections into $X$ are the subobjects of $X$. In fact, there are only set-many subobjects of $X$ (in category-theoretic language, sets form a well-powered category). It is pesky to work with set-many proper equivalence classes, so we instead look for a set $P(X)$ of injections into $X$, one from each isomorphism class. We may additionally require some nice properties, for instance, if $i : X \to Y$ is in $P(Y)$ and $j : Y \to Z$ is in $P(Y)$, we would expect $j \circ i : X \to Z$ to be in $P(Z)$. One can come up with a wish list of such nice closure conditions, here's another one: if $i : Y \to X$ and $j : Z \to X$ are in $P(X)$, and there is (a unique) $k : X \to Z$ such that $i = j \circ k$, then $k$ is in $P(Z)$.

We know the answer, of course, just take $P(X)$ to be the canonical subset inclusions into $X$. This is not the only choice of such representative inclusions, but it's a pretty good one.

We may therefore say that the canonical inclusions of subsets are "better" because they are the canonical representatives of subobjects (equivalence classes of injections).

The answer for quotient maps and surjections is dual. Consider equivalence classes of surjections, quotiented by isomorphism. There are only set-many such classes, therefore sets form a well-copowered category. (Some people say "cowell-powered" but then why not call it "ill-powered"?) This time we look for a set $Q(X)$ of surjections from $X$, each representing one equivalence class of surjections from $X$. We may take $Q(X)$ to be the set of all canonical quotient maps $X \to X/{\sim}$, or just the set of all equivalence relations on $X$. Once again, canonical quotient maps are "better" because they are the distinguished representatives of isomorphism classes of surjections.

It seems that you've got factorization of maps covered, so let me address the question of why canonical quotient maps and canonical inclusions are "better".

Given a set $X$, in general there is a proper class of injections $Y \to X$. However, many of these are isomorphic, where injections $i : Y \to X$ and $j : Z \to X$ are isomorphic if there is an isomorphism $k : Y \to Z$ such that $i = j \circ k$. The isomorphism classes of injections into $X$ are the subobjects of $X$. In fact, there are only set-many subobjects of $X$ (in category-theoretic language, sets form a well-powered category). It is pesky to work with set-many proper equivalence classes, so we instead look for a set $P(X)$ of injections into $X$, one from each isomorphism class. We may additionally require some nice properties, for instance, if $i : X \to Y$ is in $P(Y)$ and $j : Y \to Z$ is in $P(Y)$, we would expect $j \circ i : X \to Z$ to be in $P(Z)$. One can come up with a wish list of such nice closure conditions, here's another one: if $i : Y \to X$ and $j : Z \to X$ are in $P(X)$, and there is (a unique) $k : X \to Z$ such that $i = j \circ k$, then $k$ is in $P(Z)$.

We know the answer, of course, just take $P(X)$ to be the canonical subset inclusions into $X$, or just the subsets of $X$, since the canonical inclusions are determined by these. This is not the only choice of such representative inclusions, but it's a pretty good one.

We may therefore say that the canonical inclusions of subsets are "better" because they are the canonical representatives of subobjects (equivalence classes of injections).

The answer for quotient maps and surjections is dual. Consider equivalence classes of surjections, quotiented by isomorphism. There are only set-many such classes, therefore sets form a well-copowered category. (Some people say "cowell-powered" but then why not call it "ill-powered"?) This time we look for a set $Q(X)$ of surjections from $X$, each representing one equivalence class of surjections from $X$. We may take $Q(X)$ to be the set of all canonical quotient maps $X \to X/{\sim}$, or just the set of all equivalence classes on $X$. Once again, canonical quotient maps are "better" because they are the distinguished representatives of isomorphism classes of surjections.

It seems that you've got factorization of maps covered, so let me address the question of why canonical quotient maps and canonical inclusions are "better".

Given a set $X$, in general there is a proper class of injections $Y \to X$. However, many of these are isomorphic, where injections $i : Y \to X$ and $j : Z \to X$ are isomorphic if there is an isomorphism $k : Y \to Z$ such that $i = j \circ k$. The isomorphism classes of injections into $X$ are the subobjects of $X$. In fact, there are only set-many subobjects of $X$ (in category-theoretic language, sets form a well-powered category). It is pesky to work with set-many proper equivalence classes, so we instead look for a set $P(X)$ of injections into $X$, one from each isomorphism class. We may additionally require some nice properties, for instance, if $i : X \to Y$ is in $P(Y)$ and $j : Y \to Z$ is in $P(Y)$, we would expect $j \circ i : X \to Z$ to be in $P(Z)$. One can come up with a wish list of such nice closure conditions, here's another one: if $i : Y \to X$ and $j : Z \to X$ are in $P(X)$, and there is (a unique) $k : X \to Z$ such that $i = j \circ k$, then $k$ is in $P(Z)$.

We know the answer, of course, just take $P(X)$ to be the canonical subset inclusions into $X$. This is not the only choice of such representative inclusions, but it's a pretty good one.

We may therefore say that the canonical inclusions of subsets are "better" because they are the canonical representatives of subobjects (equivalence classes of injections).

The answer for quotient maps and surjections is dual. Consider equivalence classes of surjections, quotiented by isomorphism. There are only set-many such classes, therefore sets form a well-copowered category. (Some people say "cowell-powered" but then why not call it "ill-powered"?) This time we look for a set $Q(X)$ of surjections from $X$, each representing one equivalence class of surjections from $X$. We may take $Q(X)$ to be the set of all canonical quotient maps $X \to X/{\sim}$, or just the set of all equivalence classes on $X$. Once again, canonical quotient maps are "better" because they are the distinguished representatives of isomorphism classes of surjections.

It seems that you've got factorization of maps covered, so let me address the question of why canonical quotient maps and canonical inclusions are "better".

Given a set $X$, in general there is a proper class of injections $Y \to X$. However, many of these are isomorphic, where injections $i : Y \to X$ and $j : Z \to X$ are isomorphic if there is an isomorphism $k : Y \to Z$ such that $i = j \circ k$. The isomorphism classes of injections into $X$ are the subobjects of $X$. In fact, there are only set-many subobjects of $X$ (in category-theoretic language, sets form a well-powered category). It is pesky to work with set-many proper equivalence classes, so we instead look for a set $P(X)$ of injections into $X$, one from each isomorphism class. We may additionally require some nice properties, for instance, if $i : X \to Y$ is in $P(X)$ and $j : Y \to Z$ is in $P(Y)$, we would expect $j \circ i : X \to Z$ to be in $P(Z)$. One can come up with a wish-list of such nice closure conditions, here's another one: if $i : Y \to X$ and $j : Z \to X$ are in $P(X)$, and there is (a unique) $k : X \to Z$ such that $i = j \circ k$, then $k$ is in $P(Z)$.

We know the answer, of course, just take $P(X)$ to be the canonical subset inclusions into $X$, or just the subsets of $X$, since the canonical inclusions are determined by these. This is not the only choice of such representative inclusions, but it's a pretty good one.

We may therefore say that the canonical inclusions of subsets are "better" because they are the canonical representatives of subobjects (equivalence classes of injections).

The answer for quotient maps and surjections is dual. Consider equivalence classes of surjections, quotiented by isomorphism. There are only set-many such classes, therefore sets form a well-copowered category. (Some people say "cowell-powered" but then why not call it "ill-powered"?) This time we look for a set $Q(X)$ of surjections from $X$, each representing one equivalence class of surjections from $X$. We may take $Q(X)$ to be the set of all canonical quotient maps $X \to X/{\sim}$, or just the set of all equivalence classes on $X$. Once again, canonical quotient maps are "better" because they are the distinguished representatives of isomorphism classes of surjections.

It seems that you've got factorization of maps covered, so let me address the question of why canonical quotient maps and canonical inclusions are "better".

Given a set $X$, in general there is a proper class of injections $Y \to X$. However, many of these are isomorphic, where injections $i : Y \to X$ and $j : Z \to X$ are isomorphic if there is an isomorphism $k : Y \to Z$ such that $i = j \circ k$. The isomorphism classes of injections into $X$ are the subobjects of $X$. In fact, there are only set-many subobjects of $X$ (in category-theoretic language, sets form a well-powered category). It is pesky to work with set-many proper equivalence classes, so we instead look for a set $P(X)$ of injections into $X$, one from each isomorphism class. We may additionally require some nice properties, for instance, if $i : X \to Y$ is in $P(Y)$ and $j : Y \to Z$ is in $P(Y)$, we would expect $j \circ i : X \to Z$ to be in $P(Z)$. One can come up with a wish list of such nice closure conditions, here's another one: if $i : Y \to X$ and $j : Z \to X$ are in $P(X)$, and there is (a unique) $k : X \to Z$ such that $i = j \circ k$, then $k$ is in $P(Z)$.

We know the answer, of course, just take $P(X)$ to be the canonical subset inclusions into $X$, or just the subsets of $X$, since the canonical inclusions are determined by these. This is not the only choice of such representative inclusions, but it's a pretty good one.

We may therefore say that the canonical inclusions of subsets are "better" because they are the canonical representatives of subobjects (equivalence classes of injections).

The answer for quotient maps and surjections is dual. Consider equivalence classes of surjections, quotiented by isomorphism. There are only set-many such classes, therefore sets form a well-copowered category. (Some people say "cowell-powered" but then why not call it "ill-powered"?) This time we look for a set $Q(X)$ of surjections from $X$, each representing one equivalence class of surjections from $X$. We may take $Q(X)$ to be the set of all canonical quotient maps $X \to X/{\sim}$, or just the set of all equivalence classes on $X$. Once again, canonical quotient maps are "better" because they are the distinguished representatives of isomorphism classes of surjections.