Let's denote by $ \mathsf{Top}(2) $ the category of topological pairs. That is, let $ \mathsf{Top}(2) $ be the category whose objects are all the pairs $ (X,A) $ where $ X $ is a topological space [*] and $ A\subset X $, and whose maps $ f\colon (X,A)\to (Y,B) $ are the continuous functions $ f\colon X\to Y $ such that $ f(A)\subset B $.
Let's define $$ \left[(X,A),(Y,B)\right] = \left(\hom_{\mathsf{Top}(2)}((X,A),(Y,B)),\hom_{\mathsf{Top}(2)}((X,X),(Y,B))\right) $$ for every pair of pairs $ (X,A) $ and $ (Y,B) $ and $$ [f,g](\alpha) = g\circ\alpha\circ f $$ for every $ f\colon (X^\prime,A^\prime)\to (X,A) $ and $ g\colon (Y,B)\to (Y^\prime,B^\prime) $.
I've read that if we want something like the tensor-hom adjunction for $ R $-modules ($ R $ a commutative ring) to hold in $ \mathsf{Top}(2) $ (where $ \left[{-},{-}\right] $ is the right "internal" hom functor), we should define a "tensor" product between pairs as $$ (X,A)\boxplus (Y,B) = (X\times Y,{(A\times Y)}\cup {(X\times B)})\text{.} $$
In other words, given a pair $ (X,A) $, with this definition of $ (Y,B)\boxplus (X,A) $ it should be possible to prove that there is an adjunction $$ {-}\boxplus {(X,A)} \dashv \left[(X,A),{-}\right] $$ where $ {-}\boxplus {(X,A)} $ acts on $ g\colon (Y,B)\to (Y^\prime,B^\prime) $ by $ g\boxplus {(X,A)} = g\times 1_X $.
My categorical understanding of basic mathematical facts a bit shaky. Just to say, until some five minutes ago I never properly realized how the aforementioned tensor-hom adjunction that one learns in a abstract algebra course is "of a different nature" than the adjunction that defines the exponential object $ Y^X $ of $ X $ and $ Y $ in an arbitrary category [**].
Nevertheless, I would really like to learn more about tensor (monoidal) products, internal homs, closed categories, exponentials and all that, and I'm a learn-by-example kind of person, so I would really appreciate an explanation of the previous claims using those big words.
Phrased differently, I'm asking why in Algebraic Topology we define $ (X,A)\boxplus (Y,B) $ the way we do it, and in what sense this definition is dictated by a categorical imperative.
[*] More properly, suppose that all the topological spaces appearing are "nice" spaces.
[**]This pair of posts clarified something, but I am still in a phase where detailed proof and explanations are really appreciated.