Let $V_{1}, V_{2},\dots, V_{k}, W$ vector spaces over $\mathbb{F}$. We will denote:
$$\hom_{\mathbb{F}}(V_{1},V_{2},\dots, V_{k},W)=\{T:V_1\times V_2\times\dots\times V_{k} \rightarrow W\; \textrm{linear}\} $$ $$\hom_{\mathbb{F}}^{k}(V_{1},V_{2},\dots,V_{k},W)=\{\phi\in\hom_{\mathbb{F}}(V_{1},V_{2},\dots, V_{k},W)\;\textrm{k-linear}\} $$
Now, let $V_{1}, V_{2}, W$ vector spaces over $\mathbb{F}$, and consider the function $$\Omega:\hom_{\mathbb{F}}^{2}(V_{1},V_{2},W)\rightarrow \hom_{\mathbb{F}}(V_{1},\hom_{\mathbb{F}}(V_{2},W))\;\textrm{defined by}\; (\Omega(\phi)(v_{1}))(v_{2})=\phi(v_{1},v_{2}) $$ for all $\phi\in\hom_{\mathbb{F}}^{2}(V_{1},V_{2},W), v_{1}\in V_{1},v_{2}\in V_{2}.$
I need to show that $\Omega$ is well defined $($i.e., $\Omega(\phi)\in\hom_{\mathbb{F}}(V_{1},\hom_{\mathbb{F}}(V_{2},W)))$ and is an isomorphism.
I'm confused about the definition of $\Omega$, cause seems that $\Omega$ is the identity in $\hom_{\mathbb{F}}^{2}(V_{1},V_{2},W)$, not a function that assume values in $\hom_{\mathbb{F}}(V_{1},\hom_{\mathbb{F}}(V_{2},W))$. Looks like there is problems in the definition of $\Omega$. Can someone help me?
The definition $\hom_{\mathbb{F}}(V_{1},V_{2},\dots, V_{k},W)=\{T:V_1\times V_2\times\dots\times V_{k} \rightarrow W\; \textrm{linear}\}$ is superfluous and you should omit it. Actually it is nothing else than $\hom_{\mathbb{F}}(V_1\times V_2\times\dots\times V_{k}, W)$. The second definition should be corrected to $$\hom^k_{\mathbb{F}}(V_{1},V_{2},\dots, V_{k};W)=\{T:V_1\times V_2\times\dots\times V_{k} \rightarrow W\; k\textrm{-linear}\} .$$
Concerning your question:
To each $v_1 \in V_1$ we assign a map $\phi_{v_1} : V_2 \to W$ by defining $$\phi_{v_1}(v_2) = \phi(v_1,v_2) .$$ Since $\phi$ is $2$-linear, all $\phi_{v_1}$ are linear, i.e. we have $\phi_{v_1} \in \hom_{\mathbb{F}}(V_2,W)$.
The collection of the $\phi_{v_1}$, $v_1 \in V_1$, yields a map $$\Omega(\phi) : V_1 \to \hom_{\mathbb{F}}(V_2,W), \Omega(\phi)(v_1) = \phi_{v_1} .$$ It is easy to verify that $\Omega(\phi)$ is linear, i.e. that $\Omega(\phi) \in \hom_{\mathbb{F}}(V_1,\hom_{\mathbb{F}}(V_2,W))$. In fact, $\phi_{v_1+v'_1}(v_2) = \phi(v_1+v'_1,v_2) = \phi(v_1,v_2) + \phi(v'_1,v_2) = \phi_{v_1}(v_2) + \phi_{v'_1}(v_2) = (\phi_{v_1} + \phi_{v'_1})(v_2)$, hence $\Omega(\phi)(v_1+v'_1) = \phi_{v_1+v'_1} = \phi_{v_1} + \phi_{v'_1} = \Omega(\phi)(v_1) + \Omega(\phi)(v'_1)$. The equation $\Omega(\phi)(\lambda v_1) = \lambda \Omega(\phi)(v_1)$ is shown similarly.
The collection of the $\Omega(\phi)$, $\phi \in \hom^2_{\mathbb{F}}(V_1,V_2;W)$, yields a map $$\Omega : \hom^2_{\mathbb{F}}(V_1,V_2;W) \to \hom_{\mathbb{F}}(V_1,\hom_{\mathbb{F}}(V_2,W)) .$$ This can be written laconically as $(\Omega(\phi)(v_1))(v_2) = \phi(v_1,v_2)$. Again it is easy to verify that $\Omega$ is linear. It is moreover an easy (though somewhat tedious) exercise to show that $\Omega$ is an isomorphism.