<p>We show that Klemenc’s stable envelope of exact <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>∞</mi> </math></EquationSource> </InlineEquation>-categories induces an equivalence between stable <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>∞</mi> </math></EquationSource> </InlineEquation>-categories with a bounded heart structure and weakly idempotent complete exact <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>∞</mi> </math></EquationSource> </InlineEquation>-categories. Moreover, we generalise the Gillet–Waldhausen theorem to the connective algebraic K-theory of exact <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>∞</mi> </math></EquationSource> </InlineEquation>-categories and deduce a universal property of connective algebraic K-theory as an additive invariant on exact <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>∞</mi> </math></EquationSource> </InlineEquation>-categories. A key tool is a generalisation of a theorem due to Keller which provides a sufficient condition for an exact functor to induce a fully faithful functor on stable envelopes.</p>