The Stacks project

If I didn't make any mistakes, the following general argument illustrates why this is true. We will show that two definitions of being representable by stack coincide.

Note that effective descent can be easily proved by the notion of sieve and properties of 2-pullback. So what we really need to show is the following fact.

Let $j: X \to Y$ be a morphism such that the pullback by any scheme is stack. If diagonal $\Delta_Y: Y \to Y\times Y$ is representable by sheaf, then $\Delta_X$ is also representable by sheaf.

Fix $T \to X\times X$, then $u: Isom_X \to Isom_Y$ is pull back of $\Delta_{X/Y}$. Given any scheme U, the fact that $X_U$ is a stack tells us u is representable by sheaf after fibered producting with U over Y. However, the target of this morphism is sheaf, hence $Isom_X \times_Y U$ is also a sheaf. In other words, $Isom_X \to Y$ is representable by sheaf. Note that this morphism factor through sheaf $Isom_Y$, we conclude that $Isom_X$ is sheaf.

@#4911: This type of comment isn't so useful. If you think you have an essentially different argument and especially if the argument uses less machinery, then by all means suggest replacing the proof (but it has to use machinery that is already in the Stacks project of course). This is useful for everybody. But in this comment I can't even understand what "the two definitions of being representable by stack" means. Is it related to the lemma?