The category of topological spaces has a 2-category structure, in which the objects are topological spaces, morphisms are continuous maps and 2-morphisms are homotopy classes of homotopies. Therefore, we can ask what it means for two morphisms to be "adjoint" to one another. In this case, one works out that a pair of continuous maps $f: X \to Y, g: Y \to X $ are adjoint to one another if there exist homotopies $H: id_X \simeq g\circ f$ and $K: f\circ g \simeq id_Y$ such that $$f(H(x,t)) = f(x),\quad H(g(y),t)=g(y) \text{ for } x\in X,y\in Y,0\leq t\leq 1/2$$ and $$K(f(x),2t-1) = f(x),\quad g(K(y,2t-1))=g(y) \text{ for } x\in X,y\in Y,1/2\leq t\leq 1$$
Has this concept been studied? Does it go by another name? I don't have any non-trivial examples and a quick google search hasn't turned anything up, so I'm not sure whether or not it's a concept of interest.
More specifically, the 2-category structure is as follows: Let $f,g,h: X \to Y $ be continuous maps and $H: f \simeq g $, $K: g \simeq h$ be homotopies. Then the vertical composition of $H,K$ is given by $$K\circ H: I \times X \to Y$$ $$(K\circ H)(x,t) = H(x,t) \text{ if } 0\leq t\leq 1/2 $$ $$(K\circ H)(x,t) = K(x,2t-1) \text{ if } 1/2\leq t\leq 1 $$
Similarly, if $X \xrightarrow{f,f'} Y \xrightarrow{g,g'} Z$ are continuous maps and $H: f \simeq f' $, $K: g \simeq g'$ are homotopies, then the vertical composition of $H,K$ is $$K\ast H: I\times X \to Z $$ $$(K \ast H)(x,t) = K(H(x,t),t) $$
Since, we have homotopies $H: id_X \simeq g \circ f$ and $K: f \circ g \simeq id_Y$, the maps $f,g$ constitute a homotopy equivalence (in particular an adjoint equivalence). But a well-known fact about 2-categories is that any pair of objects which are equivalent are in fact adjoint equivalent. So the notion of two spaces being adjoint is the same as them being homotopy equivalent. Hence, the notion of adjointness is not very interesting.