I have proved that my category $\mathbf{Fcd}$ has small products. ~~(Correction, it seems it has no terminal object that is empty product. I have overlooked this earlier in my proof.)~~ No, it indeed has a terminal object.

Next I tried to prove that it has an exponential object (and so it cartesian closed).

I first conjectured that the exponential object $Y^X = \operatorname{id}^{\mathbf{Fcd}}_{(Y^X)}$ (where $Y^X$ in the right part of the equation means the set of graphs of functions in $Y^X$).

But if it is an exponential object, then there are isomorphisms between $Z \rightarrow \operatorname{id}^{\mathbf{Fcd}}_{( Y^X)}$ and $Z \times X \rightarrow Y$, which seems not a case.

So now I suspect that the category $\mathbf{Fcd}$ has no exponential object.

As such, how to prove that there are no exponential object in a category?

You may read about my category $\mathbf{Fcd}$ at http://www.mathematics21.org/binaries/product.pdf (to fully understand it, you need first read my book: http://www.mathematics21.org/algebraic-general-topology.html). Well, in order to be able to answer my question, you may probably don't need to read my writings first. I post links to my writings for reference.