Diagonal functor

Let $𝖩$ and $𝖢$ be categories. The corresponding diagonal functor #m/def/cat

Δ 𝖩 : 𝖢 \to 𝖢 𝖩

is the functor into the functor category $𝖢 𝖩$ sending each object $𝑋 \in 𝖢$ to the constant functor $𝑋 : 𝖩 \to 𝖢$ and each morphism $𝑓 \in 𝖢 (𝑋, 𝑌)$ to the natural transformation whose components are all $𝑓$ .

In the case $𝖩 = 𝟤 = 1 ―― \oplus 1 ――$ , we have $𝖢 𝟤 ≅ 𝖢 \times 𝖢$ , giving the typical binary diagonal functor.

Properties

The adjoints, if they exist, are limits and colimits of shape $𝖩$ in $𝖢$ — See Limits and colimits as adjoints to the diagonal

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