Definition of subgame perfect Nash equilibrium Take a two-stage game with complete information and simultaneous actions in each state: 

(1) Player 1 and 2 simultaneously choose action $a_1\in A_1$  and
$a_2\in A_2$ respectively.

(2) Player 1 and 2 observe the outcome of the 1st stage
$(a_1, a_2)$, then simultaneously choose action  $a_3\in A_3$ and
$a_4\in A_4$ respectively.

Payoffs are $u_i(a_1, a_2, a_3, a_4)$ for $i = 1,2$. 



As equilibrium concept I use subgame perfect Nash equilibrium. I find it by
backward induction: 

(A) find the functions $a^*_3(\cdot)$  and $a^*_4(\cdot)$ such that $\forall (a_1,a_2)\in A_1\times A_2$

$$
\begin{cases}
a_3^*(a_1, a_2)\in argmax_{a_3(\cdot)}u_1(a_1, a_2, a_3(a_1, a_2), a_4(a_1, a_2))\\
a_4^*(a_1, a_2)\in argmax_{a_4(\cdot)}u_2(a_1, a_2, a_3(a_1, a_2), a_4(a_1, a_2))\\
\end{cases}
$$ 

(B) find $(a_1^*, a_2^*)\in A_1\times A_2$ such that
$$
\begin{cases}
a_1^*\in argmax_{a_1}u_1(a_1, a_2, a^*_3(a_1, a_2), a^*_4(a_1, a_2))\\
a_2^*\in argmax_{a_2}u_2(a_1, a_2, a^*_3(a_1, a_2), a^*_4(a_1, a_2))\\
\end{cases}
$$ 



Question: a subgame perfect Nash equilibrium is
$$
\{a_1^*, a_2^*, \underbrace{a^*_3(\cdot), a^*_4(\cdot)}_{\text{Functions}}\}
$$
or
$$
\{a_1^*, a_2^*, \underbrace{a^*_3(a_1^*), a^*_4(a_2^*)}_{\text{Point in the image set of the functions $a^*_3(\cdot), a^*_4(\cdot)$}}\}
$$
?
An equilibrium consists of a profile of strategies, which specifies an action for every player at each possible contingency. Since each action profile $(a_1,a_2)$ is a contingency, the SPE must include functions $a^*_3(a_1,a_2), a^*_4(a_1,a_2)$ that specifies what to do at those contingencies.