Item 4 (001S) — The Clowder Project

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Interaction With Composition. Let
\[ A \underset {g}{\overset {f}{\rightrightarrows }} B \underset {k}{\overset {h}{\rightrightarrows }} C \]

be functions. We have an inclusion of sets

\[ \text{Eq}\webleft (h\circ f\circ \text{eq}\webleft (f,g\webright ),k\circ g\circ \text{eq}\webleft (f,g\webright )\webright ) \subset \text{Eq}\webleft (h\circ f,k\circ g\webright ), \]

where $\text{Eq}\webleft (h\circ f\circ \text{eq}\webleft (f,g\webright ),k\circ g\circ \text{eq}\webleft (f,g\webright )\webright )$ is the equaliser of the composition

\[ \text{Eq}\webleft (f,g\webright )\overset {\text{eq}\webleft (f,g\webright )}{\hookrightarrow }A\underset {g}{\overset {f}{\rightrightarrows }}B\underset {k}{\overset {h}{\rightrightarrows }}C. \]

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