I am trying to understand why the left-hand square of the diagram below (in a topos) is a pullback,
where $\Delta_B$ is the diagonal map, $\delta_B$ is clearly the characteristic map of $\Delta _B$ and $\langle b,1\rangle$ is the unique map induced by the universal property of the product. Clearly the right-hand square is a pullback by definition, but why is the left one? The author of the book says "the first square is a pullback by inspection". Do I have to check the very definition of a pullback involving the universal property, or is there an easier way to do this?
I have found myself before in similar situations where I have a diagram consisting of commutative squares and need to check if they are pullbacks. I wonder whether it possible to see that you actually have a pullback without a lot of calculations; if there is a general way. I know that if you have two pullbacks then you get another one by putting them together, but I am asking about situations like the above, where you just need to check whether four maps put together in a square actually form a pullback. Thanks for any help.