ct.category theory: are quotients by equivalence relations "better" than assumptions?

The nLab page you are looking for is called factoring systems. Here is my favorite, which I think answers your question. In any category with limits and finite colimitos, all morphism. $ f: X to Y $ has a canonical factorization

$$ X to text {coim} (f) to text {im} (f) to Y $$

where $ text {im} (f) $, the regular image, is the equalizer of the cokernel pair of $ f $ (this is the "nonabelian" version of "kernel of the cokernel") and $ text {coim} (f) $, the regular coimage, is the coequalizer of the kernel pair of $ f $ (again, the "nonabelian" version of "kernel cokernel"). These two constructions are categorically dual and, therefore, among other things, the image-coimage factorization of $ f $ in the opposite category is the same sequence of maps but in the opposite order.

In $ text {Set} $, the coimage and the image are the image of a function in the usual sense, but they are calculated in different ways, which I think coincide with the distinction that is obtained. $ text {coim} (f) $ is calculated, more or less, by building the equivalence relation in $ X $ defined by $ x_1 sim x_2 Leftrightarrow f (x_1) = f (x_2) $, then quoting $ X $ For this. $ text {im} (f) $ it is calculated categorically double, although at the beginning it seems a little strange: when building the pushout first $ Y sqcup_X Y $, then isolating the subset of $ Y $ of elements that are sent to the same element through both canonical maps $ Y a Y sqcup_X Y $.

In particular, the factorization you want for an injection is the normal image factorization, and the factorization you want for an injection is the normal image factorization, so in fact they are categorically dual. The full coimage image factorization combines these.

It is a nontrivial theorem that the map $ text {coim} (f) to text {im} (f) $ it's an isomorphism in $ text {Set} $. It is also an isomorphism in any Abelian category and in $ text {Grp} $ (This is an abstract form of the first isomorphism theorem), but in general it is just a monomorphism and an epimorphism. A very instructive example is $ text {Up} $, where $ text {coim} (f) $ is the image of topologized set theory as a quotient of $ X $Y $ text {im} (f) $ is the image of topologized set theory as a subspace of $ Y $. (Note that these match for compact Hausdorff spaces!)