Jonathan - thanks, fixed!

Keith - you asked a while ago if we could draw fancy diagrams here, and the answer is basically no. But yes, one can draw commutative squares! (Anyone who wants to learn how, click on the little gear to the upper right of his comment.)

By the way, the square you wrote down is not really a commutative square in any category, because \\(f\\) is not a morphism from \\(x\\) to \\(f(x)\\); it's a morphism from \\(X\\) to \\(Y\\). One should not write \\(x \stackrel{f}{\to} f(x)\\); one writes \\(f: x \mapsto f(x) \\) to indicate that the function \\(f\\) sends \\(x\\) to \\(f(x)\\).

The "mapsto" arrow, \\(\mapsto\\), has a very different meaning than the "to" arrow, \\(\to\\). For more see:

* Math Stackexchange, [Difference of mapsto and arrow.](https://math.stackexchange.com/questions/473247/difference-of-mapsto-and-right-arrow)

Keith - you asked a while ago if we could draw fancy diagrams here, and the answer is basically no. But yes, one can draw commutative squares! (Anyone who wants to learn how, click on the little gear to the upper right of his comment.)

By the way, the square you wrote down is not really a commutative square in any category, because \\(f\\) is not a morphism from \\(x\\) to \\(f(x)\\); it's a morphism from \\(X\\) to \\(Y\\). One should not write \\(x \stackrel{f}{\to} f(x)\\); one writes \\(f: x \mapsto f(x) \\) to indicate that the function \\(f\\) sends \\(x\\) to \\(f(x)\\).

The "mapsto" arrow, \\(\mapsto\\), has a very different meaning than the "to" arrow, \\(\to\\). For more see:

* Math Stackexchange, [Difference of mapsto and arrow.](https://math.stackexchange.com/questions/473247/difference-of-mapsto-and-right-arrow)