Warning 1.5.2.16. In ordinary category theory, it is sometimes useful to refer to the commutativity of diagrams in situations which do not fit the paradigm of Definition 1.5.2.5. For example, the commutativity of a diagram

\[ \xymatrix@R =50pt@C=50pt{ X \ar [r]^-{f} & Y \ar@ <.4ex>[r]^{u} \ar@ <-.4ex>[r]_{v} & Z } \]

is often understood as the requirement that $u \circ f = v \circ f$. Beware that this usage is potentially ambiguous (from the shape of the diagram alone, it is not clear that commutativity should enforce the identity $u \circ f = v \circ f$, but not the identity $u = v$), so we will take special care when applying similar terminology in the $\infty $-categorical setting.