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.