Return to Answer

I try:

Let $\mathcal{C}$ a category , by induction we build $Y^n\mathcal{C}\subset \mathcal{C}^>$ :

$Y^0\mathcal{C}$ is the image of the Yoneda immersion $h_-: \mathcal{C}\to \mathcal{C}^>$, let $Y^{n+1}\mathcal{C}$ the subcategory of $\mathcal{C}^>$ generated by $Y^{n}\mathcal{C}$ and by the ker diagrams $K \to A \rightrightarrows B $ (by $A, B\in Y^n\mathcal{C}$) and with the arrow induced by morphisms of pairs diagram in $Y^{n}\mathcal{C}$. Let $\mathcal{C}_{K}:= \bigcup_n Y^n\mathcal{C}$. Then for $F: \mathcal{C}\to \mathcal{B} $, $\mathcal{B} $ by Kernels, has for induction a extentions to a Ker-preserving funtor to any $Y^n\mathcal{C}$ (with reference to the diagram above let $F(K):= Ker(F(A) \rightrightarrows F(B)) $ and this extention in unique but isomorphisms.

Then $h_-: \mathcal{C} \to \mathcal{C}_K$ is the Ker-completions.

For duality $(h^-)^{ op }: \mathcal{C} \to ((C^{op})_K)^{ op }$ is the Coker-completion.

If above you consider $A$ and $B$ as (finite) product of objects of $Y^n\mathcal{C}$ you get the (finite)limits completion. Dually you can build the (finite)colimit completion.

Alternatively in the proof above you can consider $K$ as coker (instead Ker) of arrow pairs: $ A \rightrightarrows B \to K$, then you get $\mathcal{C}_{CK}:= \bigcup_n Y^n\mathcal{C}$ and this has the cokernel's, (or (finite)colimits taking for $A$ and $B$ the (finite) coproduct of objects of $Y^n\mathcal{C}$). But the yoneda functor $h_-: \mathcal{C}\to \mathcal{C}_{CK}$

don't preserve cokernel's , but there is advantage: the Yoneda immersion is full and dense then $\mathcal{C}_{CK}$

has the limits too (see Any cocomplete category with a dense small full subcategory is complete?).

PS. I did this proof for the first time here (no read anywhere) I hope this work, and dont get mistake.

I try:

Let $\mathcal{C}$ a category , by induction we build $Y^n\mathcal{C}\subset \mathcal{C}^>$ :

$Y^0\mathcal{C}$ is the image of the Yoneda immersion $h_-: \mathcal{C}\to \mathcal{C}^>$, let $Y^{n+1}\mathcal{C}$ the subcategory of $\mathcal{C}^>$ generated by $Y^{n}\mathcal{C}$ and by the ker diagrams $K \to A \rightrightarrows B $ (by $A, B\in Y^n\mathcal{C}$) and with the arrow induced by morphisms of pairs diagram in $Y^{n}\mathcal{C}$. Let $\mathcal{C}_{K}:= \bigcup_n Y^n\mathcal{C}$. Then for $F: \mathcal{C}\to \mathcal{B} $, $\mathcal{B} $ by Kernels, has for induction a extentions to a Ker-preserving funtor to any $Y^n\mathcal{C}$ (with reference to the diagram above let $F(K):= Ker(F(A) \rightrightarrows F(B)) $ and this extention in unique but isomorphisms.

Then $h_-: \mathcal{C} \to \mathcal{C}_K$ is the Ker-completions.

For duality $(h^-)^{ op }: \mathcal{C} \to ((C^{op})_K)^{ op }$ is the Coker-completion.

If above you consider $A$ and $B$ as (finite) product of objects of $Y^n\mathcal{C}$ you get the (finite)limits completion. Dually you can build the (finite)colimit completion.

Alternatively in the proof above you can consider $K$ as coker (instead Ker) of arrow pairs: $ A \rightrightarrows B \to K$, then you get $\mathcal{C}_{CK}:= \bigcup_n Y^n\mathcal{C}$ and this has the cokernel's, (or (finite)colimits taking for $A$ and $B$ the (finite) coproduct of objects of $Y^n\mathcal{C}$). But the yoneda functor $h_-: \mathcal{C}\to \mathcal{C}_{CK}$

don't preserve cokernel's , but there is advantage: the Yoneda immersion is full and dense then $\mathcal{C}_{CK}$

has the limits too (see Any cocomplete category with a dense small full subcategory is complete?).

PS. I did this proof for the first time here (no read anywhere) I hope this work, and dont get mistake.

I try:

Let $\mathcal{C}$ a category , by induction we build $Y^n\mathcal{C}\subset \mathcal{C}^>$ :

$Y^0\mathcal{C}$ is the image of the Yoneda immersion $h_-: \mathcal{C}\to \mathcal{C}^>$, let $Y^{n+1}\mathcal{C}$ the subcategory of $\mathcal{C}^>$ generated by $Y^{n}\mathcal{C}$ and by the ker diagrams $K \to A \rightrightarrows B $ (by $A, B\in Y^n\mathcal{C}$) and with the arrow induced by morphisms of pairs diagram in $Y^{n}\mathcal{C}$. Let $\mathcal{C}_{K}:= \bigcup_n Y^n\mathcal{C}$. Then for $F: \mathcal{C}\to \mathcal{B} $, $\mathcal{B} $ by Kernels, has for induction a extentions to a Ker-preserving funtor to any $Y^n\mathcal{C}$ (with reference to the diagram above let $F(K):= Ker(F(A) \rightrightarrows F(B)) $ and this extention in unique but isomorphisms.

Then $h_-: \mathcal{C} \to \mathcal{C}_K$ is the Ker-completions.

For duality $(h^-)^{ op }: \mathcal{C} \to ((C^{op})_K)^{ op }$ is the Coker-completion.

I try:

Let $\mathcal{C}$ a category , by induction we build $Y^n\mathcal{C}\subset \mathcal{C}^>$ :

$Y^0\mathcal{C}$ is the image of the Yoneda immersion $h_-: \mathcal{C}\to \mathcal{C}^>$, let $Y^{n+1}\mathcal{C}$ the subcategory of $\mathcal{C}^>$ generated by $Y^{n}\mathcal{C}$ and by the ker diagrams $K \to A \rightrightarrows B $ (by $A, B\in Y^n\mathcal{C}$) and with the arrow induced by morphisms of pairs diagram in $Y^{n}\mathcal{C}$. Let $\mathcal{C}_{K}:= \bigcup_n Y^n\mathcal{C}$. Then for $F: \mathcal{C}\to \mathcal{B} $, $\mathcal{B} $ by Kernels, has for induction a extentions to a Ker-preserving funtor to any $Y^n\mathcal{C}$ (with reference to the diagram above let $F(K):= Ker(F(A) \rightrightarrows F(B)) $ and this extention in unique but isomorphisms.

Then $h_-: \mathcal{C} \to \mathcal{C}_K$ is the Ker-completions.

For duality $(h^-)^{ op }: \mathcal{C} \to ((C^{op})_K)^{ op }$ is the Coker-completion.

If above you consider $A$ and $B$ as (finite) product of objects of $Y^n\mathcal{C}$ you get the (finite)limits completion. Dually you can build the (finite)colimit completion.

Alternatively in the proof above you can consider $K$ as coker (instead Ker) of arrow pairs: $ A \rightrightarrows B \to K$, then you get $\mathcal{C}_{CK}:= \bigcup_n Y^n\mathcal{C}$ and this has the cokernel's, (or (finite)colimits taking for $A$ and $B$ the (finite) coproduct of objects of $Y^n\mathcal{C}$). But the yoneda functor $h_-: \mathcal{C}\to \mathcal{C}_{CK}$

don't preserve cokernel's , but there is advantage: the Yoneda immersion is full and dense then $\mathcal{C}_{CK}$

has the limits too (see Any cocomplete category with a dense small full subcategory is complete?).

PS. I did this proof for the first time here (no read anywhere) I hope this work, and dont get mistake.

I try:

Let $\mathcal{C}$ a category , by induction we build $Y^n\mathcal{C}\subset \mathcal{C}^>$ :

$Y^0\mathcal{C}$ is the image of the Yoneda immersion $h_-: \mathcal{C}\to \mathcal{C}^>$, let $Y^{n+1}\mathcal{C}$ the subcategory of $\mathcal{C}^>$ generated by $Y^{n}\mathcal{C}$ and by the ker diagrams $K \to A \rightrightarrows B $ (by $A, B\in Y^n\mathcal{C}$) and with the arrow induced by morphisms of pairs diagram in $Y^{n}\mathcal{C}$. Let $\mathcal{C}_{K}:= \bigcup_n Y^n\mathcal{C}$. Then for $F: \mathcal{C}\to \mathcal{B} $, $\mathcal{B} $ by Kernels, has for induction a extentions to a Ker-preserving funtor to any $Y^n\mathcal{C}$ (with reference to the diagram above let $F(K):= Ker(F(A) \rightrightarrows F(B)) $ and this extention in unique but isomorphisms.

Then $Y: \mathcal{C} \to \mathcal{C}_K$ is the Ker-completions.

For duality $(h^-)^{ op }: \mathcal{C} \to (C^<)^{ op }$ is the Coker-completion.

I try:

Let $\mathcal{C}$ a category , by induction we build $Y^n\mathcal{C}\subset \mathcal{C}^>$ :

$Y^0\mathcal{C}$ is the image of the Yoneda immersion $h_-: \mathcal{C}\to \mathcal{C}^>$, let $Y^{n+1}\mathcal{C}$ the subcategory of $\mathcal{C}^>$ generated by $Y^{n}\mathcal{C}$ and by the ker diagrams $K \to A \rightrightarrows B $ (by $A, B\in Y^n\mathcal{C}$) and with the arrow induced by morphisms of pairs diagram in $Y^{n}\mathcal{C}$. Let $\mathcal{C}_{K}:= \bigcup_n Y^n\mathcal{C}$. Then for $F: \mathcal{C}\to \mathcal{B} $, $\mathcal{B} $ by Kernels, has for induction a extentions to a Ker-preserving funtor to any $Y^n\mathcal{C}$ (with reference to the diagram above let $F(K):= Ker(F(A) \rightrightarrows F(B)) $ and this extention in unique but isomorphisms.

Then $h_-: \mathcal{C} \to \mathcal{C}_K$ is the Ker-completions.

For duality $(h^-)^{ op }: \mathcal{C} \to ((C^{op})_K)^{ op }$ is the Coker-completion.