Limits and colimits as adjoints to the diagonal

Let $𝖢, 𝖩$ be categories with $𝖩$ small, and consider the diagonal functor

Δ 𝖩 : 𝖢 \to 𝖢 𝖩

from $𝖢$ to its category of diagrams. Then the left and right adjoints of $Δ 𝖩$ , if they exist are given by colimits and limits respectively.¹

l i m ⟶ 𝖩 ⊣ Δ 𝖩 ⊣ l i m ⟵ 𝖩

Proof

#missing/proof

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Footnotes