Equality of Mappings/Also defined as

Equality of Mappings
It is worth labouring the point that for two mappings to be equal, not only must their domains be equal, but so must their codomains:


 * It may seem like nit-picking to distinguish between functions which have different [ codomains but are otherwise equal] (and indeed until recently most authors did not) but failure to make the distinction sometimes leads to confusion.


 * At first sight this may appear somewhat bedantic but we shall see later that not to do so can lead to confusion.


 * This is a matter of convention. Some authors would say that $\theta$, $\phi$ are equal if $(1)$ [equality of domains] and $(3)$ [equality of images] hold, but this gives a little difficulty when defining the inverse mapping.
 * -- : $\S 3.2$. Equality of mappings (footnote)

Thus it is worth noting that this is a modern departure, and many earlier sources, for example, do not impose this condition, stating merely that two mappings are equal if the domains are equal and every element of the domain has the same image under each mapping.

Other sources, for example and, gloss over the topic, and merely assume equality of domains and codomains, implicitly stating the equality of the images.

, while careful to state equality of codomains, implicitly assumes that two mappings in question have the same domain to start with, and so misses a chance to explain the equality of the domains:
 * ... It also follows that two maps $f: X \to Y$ and $g: X \to Z$ can only be equal if $Y = Z$.

Earlier works, for example and  tend to identify a mapping with its graph, preferring to dispense with a need to specify either its domain or its codomain.