Pointwise topology embedding First let $\Lambda$ be the bijective mapping between $Y^{Z \times X}$ and $(Y^X)^Z$ defined as follows: every mapping $f: Z \times X \to Y$ defines a set of mappings from $X$ to $Y$: for each $z \in Z$ is $f_z:X \to Y$ defined as $f_z(x) = f(z,x)$. The mapping $z \mapsto f_z$ of $Z$ to $Y^X$ obtained this way we denote as $\Lambda(f)$.
Engelking calls this the exponential mapping.
Then he goes on to define a topology on $C(X, Y)$ called the pointwise topology as the restriction of the product topology on $Y^X$ restricted to $C(X,Y)$. We can also see that this is equal to the topology generated by the subbasis
$$\{M(x, U) : x \in X \text{ and $U$ open in $Y$}\},$$
where $M(x,U) := \{f \in C(X,Y) : f(x) \in U\}$.
So now I can finally ask the question I want to ask...
Give $C(X,Y)$, $C(Z \times X, Y)$ and $C(Z, C(X, Y))$ the pointwise topology. 
How do I now show that $\Lambda:C(Z \times X, Y) \to C(Z, C(X,Y))$ is an embedding? I'm drowning in a syntax mess. I don't need a full solution a road map is fine.
 A: I'll show continuity. The pointwise topology on $C(X,Y)$ is precisely what it says: a net $g_i$ converges in it, $g_i\to g$, if and only if for all $x\in X$ we have $g_i(x)\to g(x)$ in $Y$. (Indeed, this is because a net in a product converges iff all projections converge.)
So suppose $f_i\to f$ in $C(Z\times X,Y)$. We need to show $\Lambda (f_i)\to \Lambda (f)$. The latter means that, for all $z\in Z$, we have $(f_i)_z\to f_z$ in $C(X,Y)$ . But this means that for all $z\in Z$, we have that for all $x\in X$, $(f_i)_z(x)\to f_z(x)$ in $Y$. In other words, it means that for all $(z,x)\in Z\times X$, we have $f_i(z,x)\to f(z,x)$ in $Y$. But this is precisely our assumption $f_i\to f$ in $C(Z\times X,Y)$.
So it's basically a tautology: the maps are defined by 'currying' (holding the function pointwise fixed), and we are considering the pointwise topology.
To show that in fact $\Lambda(f)$ is continuous for continuous $f$, a similar argument can be used (namely $\Lambda(f)(z_i)\to \Lambda(f)(z)$ for nets $z_i\to z$), and the same for the well-definedness and continuity of the inverse.
(I think this is one of the places where using nets is superior to working directly with open set / subbases of the topology.)
A: For the pointwise topology, I don't know. For the compact-open topology, with hypotheses on the spaces, the map you are talking about can be a homeomorphism and you can find it in the book of Maunder. For the pointwise topology, there are results in the book of Munkres.
