Item 4 (002W) — The Clowder Project

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Style 1	`book`	Alegreya Sans	`tcbthm`
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*To be replaced with Linus Romer's Elemaints when it is released.

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

be functions. We have a surjection

\[ \text{CoEq}\webleft (h\circ f,k\circ g\webright )\twoheadrightarrow \text{CoEq}\webleft (\text{coeq}\webleft (h,k\webright )\circ h\circ f,\text{coeq}\webleft (h,k\webright )\circ k\circ g\webright ) \]

exhibiting $\text{CoEq}\webleft (\text{coeq}\webleft (h,k\webright )\circ h\circ f,\text{coeq}\webleft (h,k\webright )\circ k\circ g\webright )$ as a quotient of $\text{CoEq}\webleft (h\circ f,k\circ g\webright )$ by the relation generated by declaring $h\webleft (y\webright )\sim k\webleft (y\webright )$ for each $y\in B$.

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