Category:Morphism Categories
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This category contains results about Morphism Categories.
Definitions specific to this category can be found in Definitions/Morphism Categories.
Let $\mathbf C$ be a metacategory.
Its morphism category, denoted $\mathbf C^\to$, is defined as follows:
| Objects: | The morphisms $\mathbf C_1$ of $\mathbf C$ | |
| Morphisms: | $g: f \to f'$ is a pair $\tuple {g_1, g_2}$ of morphisms of $\mathbf C$ such that $g_2 \circ f = f' \circ g_1$ | |
| Composition: | $\tuple {h_1, h_2} \circ \tuple {g_1, g_2} := \tuple {h_1 \circ g_1, h_2 \circ g_2}$, whenever this is defined | |
| Identity morphisms: | $\operatorname{id}_f := \tuple {\operatorname{id}_C, \operatorname{id}_D}$ for $f: C \to D$ |
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