Category-vector space analogy

The is a striking, albeit informal, correspondence between the basic concepts of category theory and linear algebra. The following table, based on 2011. Introduction to representation theory, §7.6, is a summary.

Category theory	Linear algebra
category $𝖢$	vector space $𝑉$
hom-set $𝖢 (𝑋, 𝑌)$	nondegenerate bilinear form $⟨ 𝑥, 𝑦 ⟩$
opposite category	opposite inner product
Category of sets	ground field $𝕂$
full subcategory	nondegenerate subspace
functor $𝐹 : 𝖢 \to 𝖣$	linear map $𝑓 : 𝑉 \to 𝑊$
presheaf	linear form
Yoneda lemma	nondegeneracy
representable functor	representable form
adjoint functor	adjoint operator