In combinatorics, one considers both maps out of a space $X$ (colourings) and maps into a space $X$ (tuples). But there is one key difference between the two: if $X$ has $n$ elements, then the number of maps from, say, $\{0,1\}$ to $X$ is polynomial in $n$ (it has order $n^2$), while the number of maps from $X$ to $\{0,1\}$ is exponential in $n$ (it has order $2^n$). Thus we see that maps out of $X$ into a simple space form a much larger, and presumably thus much richer, space than maps into $X$ from a simple space. For instance, deciding whether a four-colouring of a graph with certain specified properties exists is usually a harder problem than deciding whether a four-tuple in a graph with certain specified properties exists, although both questions can be interesting.

Of course, the situation could be different for other categories than the combinatorial one (particularly if some sort of duality is available)...