Definition 6.1. Let $M_{j} = (M_{j}, I_{j})$ be $L$ -structures for each $j \in J$ . Let $U$ be an ultraproduct on $J$ . Define an interpretation $I_{U}$ of $L$ on $\prod_{j \in J} M_{j} ∕ U$ . Let $S \in L$ .

If $S$ is an $n$ -ary relation:
$I_{U} (S) = [{(I_{j} (S))}_{j \in J}] \subseteq {(\prod_{j \in J} M_{j} ∕ U)}^{n} .$
If $S$ is a constant:
$I_{U} (S) = [{(I_{j} (S))}_{j \in J}] \in \prod_{j \in J} M ∕ U .$
Functions are a bit less clear. However, we can always turn a function into a relation by looking at its graph (i.e. $f : M^{n} \to M$ has graph $R_{f} = {(\bar{x}, y) \in M^{n + 1} : f (\bar{x}) = y}$ ).

So if $S$ is a function: for each $j \in J$ , define the graph of $I_{j} (S)$ as
$G_{j} (S) = {(a_{1}, \dots, a_{n}, b) \in M_{j}^{n + 1} : I_{j} (S) (a_{1}, \dots, a_{n}) = b} .$

Then $[{(G_{j} (S))}_{j \in J}]$ is the graph of a function
${(\prod_{j \in J} M_{j} ∕ U)}^{n} \to \prod_{j \in J} M_{j} ∕ U .$

(Checking this is left as an exercise). Now define $I_{U} (S)$ to be the function corresponding to $[{(G_{j} (S))}_{j \in J}]$ .