How are equality of morphisms described in a category? So, a category is made of objects and morphisms with some axioms on the morphism. I want to ask, what exactly does it mean for two morphisms to be equal? (eg: given in associatvity property in a category)
 A: This question is actually trickier than it seems! The foundations-agnostic answer is that every category comes equipped with an equivalence relation on its morphisms, which composition must respect — what it means for two morphisms to be equal is something we must decide when constructing a category.
When working in traditional, set-theoretic foundations (i.e. what 99% of mathematicians are implicitly doing), every category has a canonical choice of equivalence for morphisms: Since $f, g \in \mathrm{Hom}(A,B)$ and $\mathrm{Hom}$ is a set, we can ask whether $f = g$.
In different foundations, we must explicitly keep track of how morphisms are being identified. For example, in weak type theories (e.g. intensional Martin-Löf type theory without function extensionality), what is internally called a "category" is what a traditional mathematician would call an $\mathscr{E}$-category, as in the paper "Category theoretic structure of setoids" by Kinoshita and Power (link on ScienceDirect — note: an $\mathscr{E}$-category is a category enriched over the ex/lex completion of Sets). Explicitly, the data of such a category consists of

*

*A type of objects,

*For each pair of objects $A, B$, a type $\mathrm{Hom}(A, B)$,

*For each $A, B$, an equivalence relation on the type $\mathrm{Hom}(A, B)$

*The usual identities and compositions,

*Such that the composition map respects $\approx$, i.e. $f \approx f'$ and $g \approx g'$ imply $(f \circ g) \approx (f' \circ g')$.

The same approach would be (implicitly) applied if formalising category theory within the context of Bishop's constructive mathematics, since there, a "set" by definition comes equipped with an equivalence relation (as opposed to "pre-sets", which are just stuff), and "functions" must respect this relation (as opposed to "operations", which are just mappings of stuff).
