Variant 11.6.0.130 (Strictly Unitary $2$-Categories). The contents of this tag can be found in Definition 2.2.7.1.

We say that a $2$-category $\operatorname{\mathcal{C}}$ is *strictly unitary* if, for every $1$-morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$, we have equalities

\[ \operatorname{id}_{Y} \circ f = f = f \circ \operatorname{id}_{X}, \]

and the left and right unit constraints $\lambda _{f}$, $\rho _{f}$ are the identity $2$-morphisms from $f$ to itself. Every strict $2$-category is strictly unitary, but the converse is false: we will see later that every $2$-category is isomorphic (in an appropriate sense) to a strictly unitary $2$-category (see Example 11.6.0.134).