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.