In book: https://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf chapter: "4.4.4 Categories enriched in a symmetric monoidal category" declared:
categories should really be called Set-categories
My understand:
morphisms between $ a, b $ in Category are presented by set(morphism) elements in Set-category
$ id_x $ in Category is presented by morphism: $ I \to \mathscr{X}(x, x) $ in Set-category
But I don't know:
how Set-category present $ g \circ f \in \mathscr{C}(a, c) $ where $ f:a \to b, g: b \to c $ in Catgegory?
how Set-category present unitality and associativity in Category?
By definition, a category $C$ enriched in a monoidal category $V$ is a class $\mathrm{Ob}(C)$ together with an object $C(a,b)$ of $V$ for every $a,b,\in\mathrm{Ob}(C)$ and morphisms $C(b,c)\otimes C(a,b)\to C(a,c)$ for every $a,b,c\in\mathrm{Ob}(C)$ and $I\to C(a,a)$ for every $a$ such that the two ways of combining these morphisms to get from $C(c,d)\otimes \left(C(b,c)\otimes C(a,b)\right)$ to $C(a,d)$ are equal and such that the induced morphism $C(a,b)\cong I\otimes C(a,b)\to C(b,b)\otimes C(a,b)\to C(a,b)$ and its right-handed analogue are the identity of $C(a,b)$.
If you set $V$ to be Set, then this is simply the word-for-word definition of a locally small category (many authors assume all categories to be locally small,) with the associativity and unitality constraints stated in terms of the composition functions rather than their values on individual morphisms.
More specifically: a locally small category is a class $\mathrm{Ob}(C)$ together with a set $C(a,b)$ for every $a,b\in \mathrm{Ob}(C)$ and functions $C(b,c)\times C(a,b)\to C(a,c)$ (mapping $(g,f)\mapsto g\circ f$) and $\{*\}\to C(a,a)$ (mapping $*$ to $\mathrm{id}_a$) such that the two ways of combining the composition functions to compose three morphisms are equal (the two ways being those that send $(f,g,h)\in C(c,d)\times C(b,c)\times C(a,b))$ to $f\circ(g\circ h)$ and $(f\circ g)\circ h$) and the induced function $C(a,b)\cong \{*\}\times C(a,b)\to C(b,b)\times C(a,b)\to C(b,b)$ (sending $f$ to $(*,f)$ to $(\mathrm{id}_b,f)$ to $\mathrm{id}_b\circ f$) and its right-handed analogue are the identity functions.