Definition 2.1.1.1. Let $\operatorname{\mathcal{C}}$ be a category. A nonunital strict monoidal structure on $\operatorname{\mathcal{C}}$ is a functor
which is strictly associative in the following sense:
For every triple of objects $X,Y,Z \in \operatorname{\mathcal{C}}$, we have an equality $X \otimes (Y \otimes Z) = (X \otimes Y) \otimes Z$ (as objects of $\operatorname{\mathcal{C}}$).
For every triple of morphisms $f: X \rightarrow X'$, $g: Y \rightarrow Y'$, $h: Z \rightarrow Z'$, we have an equality
\[ f \otimes (g \otimes h) = (f \otimes g) \otimes h \]of morphisms in $\operatorname{\mathcal{C}}$ from the object $X \otimes (Y \otimes Z) = (X \otimes Y) \otimes Z$ to the object $X' \otimes (Y' \otimes Z') = (X' \otimes Y') \otimes Z'$.
A nonunital strict monoidal category is a pair $(\operatorname{\mathcal{C}}, \otimes )$, where $\operatorname{\mathcal{C}}$ is a category and $\otimes : \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ is a nonunital strict monoidal structure on $\operatorname{\mathcal{C}}$.