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Since the subspace topology is a special case of an initial topology, and you are asking about generalizations anyway, I'm going to describe how the initial topology can be expressed categorically.

Let $U:\mathbf{Top}\to\mathbf{Set}$ be the forgetful functor and let $(X_i)_I$ be a family of spaces and also, by abuse of notation, the corresponding functor form the indexing set $I$ to $\mathbf{Top}$. If we want to equip a set $S$ with the initial topology with respect to some family of functions $(f_i:S\to X_i)_I$, then what we actually want is a cone $(f_i:(S,\tau_S)\to (X_i,\tau_i))$ over $(X_i)_I$ in $\mathbf{Top}$ such that $\text{Id}:(Uf_i:U(S,\tau_S)\to U(X_i,\tau_i))_I\longrightarrow(f_i:S\to X_i)_I$ is a universal arrow from $U$ to the cone $(f_i:S\to X_i)_I$.
Here, universal means that for each cone of spaces $(g_i:(Y,\tau_Y)\to(X_i,\tau_i))_I$ and each arrow from $(Ug_i:U(Y,\tau_Y)\to U(X_i,\tau_i))_I$ to $(f_i:S\to X_i)_I$, which is the same as a set map $h:Y\to S$ such that $f_i\circ h=g_i$, there is exactly one continuous map $h':(Y,\tau_Y)\to(S,\tau_S)$ satisfying $f_i\circ h'=g_i$ (formally it is a map between cones) such that $\text{Id}\circ Uh'=h$. Now, this simply means that $h'=h$ and that the set map $h$ is actually a continuous map.

Replacing $\mathbf{Top}$ and $\mathbf{Set}$ by arbitrary topologies gives the notion of a strictly initial lift. We can generalize further by allowing the $\text{Id}$ in the above description to be an isomorphism (initial lift) or even a morphism (semi initial lift).

Have a look at this nLab article which describes the dual notion of a final lift.

There is also section V.9 Adjoints in Topology in Mac Lane's Categories for the Working Mathematician, which deals with this construction and how it can be used to construct the limits in $\mathbf{Top}$.

Since the subspace topology is a special case of an initial topology, and you are asking about generalizations anyway, I'm going to describe how the initial topology can be expressed categorically.

Let $U:\mathbf{Top}\to\mathbf{Set}$ be the forgetful functor and let $(X_i)_I$ be a family of spaces and also, by abuse of notation, the corresponding functor from the indexing set $I$ to $\mathbf{Top}$. If we want to equip a set $S$ with the initial topology with respect to some family of functions $(f_i:S\to X_i)_I$, then what we actually want is a cone $(f_i:(S,\tau_S)\to (X_i,\tau_i))$ over $(X_i)_I$ in $\mathbf{Top}$ such that $\text{Id}:(Uf_i:U(S,\tau_S)\to U(X_i,\tau_i))_I\longrightarrow(f_i:S\to X_i)_I$ is a universal arrow from $U$ to the cone $(f_i:S\to X_i)_I$.
Here, universal means that for each cone of spaces $(g_i:(Y,\tau_Y)\to(X_i,\tau_i))_I$ and each arrow from $(Ug_i:U(Y,\tau_Y)\to U(X_i,\tau_i))_I$ to $(f_i:S\to X_i)_I$, which is the same as a set map $h:Y\to S$ such that $f_i\circ h=g_i$, there is exactly one continuous map $h':(Y,\tau_Y)\to(S,\tau_S)$ satisfying $f_i\circ h'=g_i$ (formally it is a map between cones) such that $\text{Id}\circ Uh'=h$. Now, this simply means that $h'=h$ and that the set map $h$ is actually a continuous map.

Replacing $\mathbf{Top}$ and $\mathbf{Set}$ by arbitrary categories gives the notion of a strictly initial lift. We can generalize further by allowing the $\text{Id}$ in the above description to be an isomorphism (initial lift) or even just a morphism (semi initial lift).

Have a look at this nLab article which describes the dual notion of a final lift.

There is also section V.9 Adjoints in Topology in Mac Lane's Categories for the Working Mathematician, which deals with this construction and how it can be used to construct the limits in $\mathbf{Top}$.

Since the subspace topology is a special case of an initial topology, and you are asking about generalizations anyway, I'm going to describe how the initial topology can be expressed categorically.

Let $U:\mathbf{Top}\to\mathbf{Set}$ be the forgetful functor and let $(X_i)_I$ be a family of spaces, and also by abuse of notation, the corresponding functor form the indexing set $I$ to $\mathbf{Top}$. If we want to equip a set $S$ with the initial topology with respect to some family of functions $(f_i:S\to X_i)_I$, then what we actually want is a cone $(f_i:(S,\tau_S)\to (X_i,\tau_i))$ over $(X_i)_I$ such that $\text{Id}:(Uf_i:U(S,\tau_S)\to U(X_i,\tau_i))\longrightarrow(f_i:S\to X_i)$ is a universal arrow from $U$ to the cone $(f_i:S\to X_i)$.
That means that for each cone of spaces $(g_i:(Y,\tau_Y)\to(X,\tau_X))$ and each arrow from $(Ug_i:U(Y,\tau_Y)\to U(X_i,\tau_i))$ to $(f_i:S\to X_i)$, which is the same as a set map $h:Y\to S$ such that $f_i\circ h=g_i$, there is exactly one continuous map $h':(Y,\tau_Y)\to(S,\tau_S)$ satisfying $f_i\circ h'=g_i$ (formally it is a map between cones) such that $\text{Id}\circ Uh'=h$. Now, this simply means that $h'=h$ and the set map $h$ is actually a continuous map.

Replacing $\mathbf{Top}$ and $\mathbf{Set}$ by arbitrary topologies gives the notion of a strictly initial lift. We can generalize further by allowing the $\text{Id}$ in the above description to be any morphism (semi initial lift) or any isomorphism (initial lift).

Have a look at this nLab article which describes the dual notion of a final lift.

Since the subspace topology is a special case of an initial topology, and you are asking about generalizations anyway, I'm going to describe how the initial topology can be expressed categorically.

Let $U:\mathbf{Top}\to\mathbf{Set}$ be the forgetful functor and let $(X_i)_I$ be a family of spaces and also, by abuse of notation, the corresponding functor form the indexing set $I$ to $\mathbf{Top}$. If we want to equip a set $S$ with the initial topology with respect to some family of functions $(f_i:S\to X_i)_I$, then what we actually want is a cone $(f_i:(S,\tau_S)\to (X_i,\tau_i))$ over $(X_i)_I$ in $\mathbf{Top}$ such that $\text{Id}:(Uf_i:U(S,\tau_S)\to U(X_i,\tau_i))_I\longrightarrow(f_i:S\to X_i)_I$ is a universal arrow from $U$ to the cone $(f_i:S\to X_i)_I$.
Here, universal means that for each cone of spaces $(g_i:(Y,\tau_Y)\to(X_i,\tau_i))_I$ and each arrow from $(Ug_i:U(Y,\tau_Y)\to U(X_i,\tau_i))_I$ to $(f_i:S\to X_i)_I$, which is the same as a set map $h:Y\to S$ such that $f_i\circ h=g_i$, there is exactly one continuous map $h':(Y,\tau_Y)\to(S,\tau_S)$ satisfying $f_i\circ h'=g_i$ (formally it is a map between cones) such that $\text{Id}\circ Uh'=h$. Now, this simply means that $h'=h$ and that the set map $h$ is actually a continuous map.

Replacing $\mathbf{Top}$ and $\mathbf{Set}$ by arbitrary topologies gives the notion of a strictly initial lift. We can generalize further by allowing the $\text{Id}$ in the above description to be an isomorphism (initial lift) or even a morphism (semi initial lift).

Have a look at this nLab article which describes the dual notion of a final lift.

There is also section V.9 Adjoints in Topology in Mac Lane's Categories for the Working Mathematician, which deals with this construction and how it can be used to construct the limits in $\mathbf{Top}$.

Since the subspace topology is a special case of an initial topology, and you are asking about generalizations anyway, I'm going to describe how the initial topology can be expressed categorically.

Let $U:\mathbf{Top}\to\mathbf{Set}$ be the forgetful functor. If we want to equip a set $S$ with the initial topology with respect to some family of functions $(f_i:S\to X_i)_I$, then what we actually want is a cone $(f_i:(S,\tau_S)\to (X_i,\tau_i))$ such that $\text{Id}:(Uf_i:U(S,\tau_S)\to U(X_i,\tau_i))\longrightarrow(f_i:S\to X_i)$ is a universal arrow from $U$ to the cone $(f_i:S\to X_i)$.
That means that for each cone of spaces $(g_i:(Y,\tau_Y)\to(X,\tau_X))$ and each arrow from $(Ug_i:U(Y,\tau_Y)\to U(X_i,\tau_i))$ to $(f_i:S\to X_i)$, which is the same as a set map $h:Y\to S$ such that $f_i\circ h=g_i$, there is exactly one continuous map $h':(Y,\tau_Y)\to(S,\tau_S)$ satisfying $f_i\circ h'=g_i$ (formally it is a map between cones) such that $Uh'=h$. Now, this simply means that $h'=h$ and the set map $h$ is actually a continuous map.

Replacing $\mathbf{Top}$ and $\mathbf{Set}$ by arbitrary topologies gives the notion of a strictly initial lift. We can generalize further by allowing the $\text{Id}$ in the above description to be any morphism (semi initial lift) or any isomorphism (initial lift).

Have a look at this nLab article which describes the dual notion of a final lift.

Since the subspace topology is a special case of an initial topology, and you are asking about generalizations anyway, I'm going to describe how the initial topology can be expressed categorically.

Let $U:\mathbf{Top}\to\mathbf{Set}$ be the forgetful functor and let $(X_i)_I$ be a family of spaces, and also by abuse of notation, the corresponding functor form the indexing set $I$ to $\mathbf{Top}$. If we want to equip a set $S$ with the initial topology with respect to some family of functions $(f_i:S\to X_i)_I$, then what we actually want is a cone $(f_i:(S,\tau_S)\to (X_i,\tau_i))$ over $(X_i)_I$ such that $\text{Id}:(Uf_i:U(S,\tau_S)\to U(X_i,\tau_i))\longrightarrow(f_i:S\to X_i)$ is a universal arrow from $U$ to the cone $(f_i:S\to X_i)$.
That means that for each cone of spaces $(g_i:(Y,\tau_Y)\to(X,\tau_X))$ and each arrow from $(Ug_i:U(Y,\tau_Y)\to U(X_i,\tau_i))$ to $(f_i:S\to X_i)$, which is the same as a set map $h:Y\to S$ such that $f_i\circ h=g_i$, there is exactly one continuous map $h':(Y,\tau_Y)\to(S,\tau_S)$ satisfying $f_i\circ h'=g_i$ (formally it is a map between cones) such that $\text{Id}\circ Uh'=h$. Now, this simply means that $h'=h$ and the set map $h$ is actually a continuous map.

Replacing $\mathbf{Top}$ and $\mathbf{Set}$ by arbitrary topologies gives the notion of a strictly initial lift. We can generalize further by allowing the $\text{Id}$ in the above description to be any morphism (semi initial lift) or any isomorphism (initial lift).

Have a look at this nLab article which describes the dual notion of a final lift.