Definition 0.1   A (small) category $\mathcal{C}$ consists of a set of objects $C_0$ and a set of arrows $C_1$ together with the following structure:

• a source map $s: C_1 \to C_0$ assigning an object $s(f)$ to each arrow $f \in C_1$ ,
• a target map: $t: C_1 \to C_0$ assigning an object t(f) to each arrow $f \in C_1$ ,
• an identity map $1 : C_0 \to C_1$ assigning to each object $A$ an arrow $1_A$ with $$s(1_A) = t(1_A) = A,$$
• a composition map $\circ : C_1 \times C_1 \to C_1$ assigning to each pair of arrows $f,g$ , such that $$s(g) = t(f)$$ , a third arrow $g \circ f$ with $s(g \circ f) = s(f)$ and $t(g \circ f) = t(g)$ .
• The composition thus defined ``$\circ$ '' is associative, that is, $$ h \circ (g \circ f) = (h \circ g) \circ f$$ whenever these compositions make sense.
• the identity map satisfies $f \circ 1_A = f $ for any $f$ such that $s(f) = A$ and $1_A \circ g = g$ , and any $g$ such that $ t(g) = A$ .