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The symbol $\cong$ denotes an isomorphism in the category of sets, which means it is simply a bijection between the two hom-sets. Because hom-sets are sets, their isomorphisms are bijective functions.

Conditions: The compared objects are hom-sets; The ambient category for the isomorphism is the category of sets

The symbol $\cong$ denotes an isomorphism in the category of sets, which means it is simply a bijection between the two hom-sets. Because hom-sets are sets, their isomorphisms are bijective functions.

Conditions: The compared objects are hom-sets; The ambient category for the isomorphism is the category of sets

For functors $F:\mathcal{A}\to\mathcal{B}$ and $G:\mathcal{B}\to\mathcal{A}$, $F$ is left adjoint to $G$ (written $F \dashv G$) if there is a natural isomorphism between the hom-set $\mathcal{B}(F(A),B)$ and the hom-set $\mathcal{A}(A,G(B))$ for all objects $A \in \mathcal{A}$ and $B \in \mathcal{B}$. The video emphasizes that an adjunction is a chosen natural isomorphism, not just an abstract existence claim.

Conditions: $F$ is a functor $\mathcal{A} \to \mathcal{B}$; $G$ is a functor $\mathcal{B} \to \mathcal{A}$; $A$ is an object of $\mathcal{A}$; $B$ is an object of $\mathcal{B}$; The isomorphism is natural in both $A$ and $B$

For functors $F:\mathcal{A}\to\mathcal{B}$ and $G:\mathcal{B}\to\mathcal{A}$, $F$ is left adjoint to $G$ (written $F \dashv G$) if there is a natural isomorphism between the hom-set $\mathcal{B}(F(A),B)$ and the hom-set $\mathcal{A}(A,G(B))$ for all objects $A \in \mathcal{A}$ and $B \in \mathcal{B}$. The video emphasizes that an adjunction is a chosen natural isomorphism, not just an abstract existence claim.

Conditions: $F$ is a functor $\mathcal{A} \to \mathcal{B}$; $G$ is a functor $\mathcal{B} \to \mathcal{A}$; $A$ is an object of $\mathcal{A}$; $B$ is an object of $\mathcal{B}$; The isomorphism is natural in both $A$ and $B$