Enriched completeness is cotensors plus small conical limits

If $\mathcal{B}$ is enriched complete, then it has every enriched weighted limit by [L3]. In particular it has the one-object weights of [L1] and the constant-unit weights of [L2], so it has all cotensors and all small conical limits.

Conversely, assume $\mathcal{B}$ has all cotensors and all small conical limits. For a small weight $W:\mathcal{A}\to \mathcal{V}$ and diagram $T:\mathcal{A}\to \mathcal{B}$, form the enriched end ${\int }_{A}W(A)⋔T(A)\mathrm{.}$ Its standard equalizer presentation uses a small product of the displayed cotensors and a small product of cotensors encoding the two action maps for every ordered pair of objects of $\mathcal{A}$. All those cotensors exist by hypothesis, and the products and equalizer are small conical limits, so the end exists. Applying $\mathcal{B}(B,-)$ and the cotensor identities identifies this end with $[\mathcal{A},\mathcal{V}](W,\mathcal{B}(B,T-)),$ which is precisely the weighted-limit universal property of [L3].

Therefore enriched completeness is equivalent to the joint existence of cotensors and all small conical enriched limits.