Let consider the subcategory $kTop \subset Top$ of compactly generated spaces. Obviously we have two functors:
$j: kTop \to Top $ , the canonical embedding functor and
$k: Top \to kTop, T \mapsto k(T)$ , which endows $T$ with topology of compactly generated spaces.
Here for topology of compactly generated spaces is defined as follows:
$U \subset X$ open if and only if $s^{-1}U \subset K$ open for all locally compact spaces $K$ and continuous $s:K \to X$.
My question is: Why is $k$ right adjoint to $j$,so why for all $X \in kTop$ and $Y \in Top$ we have bijection $Hom_{Top}(j(X), Y) \cong Hom_{kTop}(X, k(Y)) $?