Definition 4.2.1. A *category* $\mathcal{C}$ consists of the following data:

A set of objects $\mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$.

For each pair $x, y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ a set of morphisms $\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(x, y)$.

For each triple $x, y, z\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ a composition map $ \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(y, z) \times \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(x, y) \to \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(x, z) $, denoted $(\phi , \psi ) \mapsto \phi \circ \psi $.

These data are to satisfy the following rules:

For every element $x\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ there exists a morphism $\text{id}_ x\in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(x, x)$ such that $\text{id}_ x \circ \phi = \phi $ and $\psi \circ \text{id}_ x = \psi $ whenever these compositions make sense.

Composition is associative, i.e., $(\phi \circ \psi ) \circ \chi = \phi \circ ( \psi \circ \chi )$ whenever these compositions make sense.

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