= Solution
The natural-number successor <chain> embeds into the <integer> successor <chain> by ordinary inclusion. Thus $\mathbb Z$ witnesses the existential joint satisfiability in the first assertion. The two <chains> are not <elementarily equivalent>: every <integer> has a predecessor, whereas zero has none in the natural-number <chain>.
For the universal assertion, construct an <elementary extension> of the natural-number <chain> containing a two-sided successor <chain>. To its <elementary diagram> add distinct constants $d_j$ for $j\in\mathbb Z$ with $s(d_j)=d_{j+1}$ and require them to differ from all named <natural numbers>. Every finite <subset> is realized in the natural-number <chain>: place the finitely many required consecutive blocks sufficiently far to the right, away from the finitely many forbidden named <Elements>. Compactness yields an <elementary extension> $A^*$ with the required two-sided <chain>. That <chain> is a <substructure of a first-order structure>[substructure] isomorphic to the <integer> successor structure and hence is a <first-order model>[model] of its <complete theory>.
Consequently both joint <first-order theory>[theories] in the two assertions are satisfiable, though the <integer> <chain> cannot embed into the natural-number <chain> itself: repeated predecessors would give an infinite descending <sequence> of <natural numbers>. The second assertion correctly uses an <elementarily equivalent> ambient structure rather than the original one.
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