Let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123615004577&_mathId=si1.gif&_user=111111111&_pii=S0022123615004577&_rdoc=1&_issn=00221236&md5=dfad4a7fb843a20103aa10b5be5e0e1e" title="Click to view the MathML source">Kϑclass="mathContainer hidden">class="mathCode"> be a model space generated by an inner function ϑ . We study the Schatten class membership of composition operators class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123615004577&_mathId=si2.gif&_user=111111111&_pii=S0022123615004577&_rdoc=1&_issn=00221236&md5=63548c6968764d51449dce89ded5e18a" title="Click to view the MathML source">Cφ:Kϑ→H2(D)class="mathContainer hidden">class="mathCode"> with a holomorphic function class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123615004577&_mathId=si20.gif&_user=111111111&_pii=S0022123615004577&_rdoc=1&_issn=00221236&md5=d6efdfd90d4bc23731c5bd94b05f40ff" title="Click to view the MathML source">φ:D→Dclass="mathContainer hidden">class="mathCode">, and, more generally, of embeddings class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123615004577&_mathId=si4.gif&_user=111111111&_pii=S0022123615004577&_rdoc=1&_issn=00221236&md5=7d6c1a10cbd314dff7843a5a57d0c5b4" title="Click to view the MathML source">Iμ:Kθ→L2(μ)class="mathContainer hidden">class="mathCode"> with a positive measure μ in class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123615004577&_mathId=si5.gif&_user=111111111&_pii=S0022123615004577&_rdoc=1&_issn=00221236&md5=b896bfc72807ecc3895edffa3fcc155b">class="imgLazyJSB inlineImage" height="15" width="14" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022123615004577-si5.gif">class="mathContainer hidden">class="mathCode">. In the case of one-component inner functions ϑ we show that the problem can be reduced to the study of natural extensions of I and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123615004577&_mathId=si6.gif&_user=111111111&_pii=S0022123615004577&_rdoc=1&_issn=00221236&md5=6dc8dc4c25f403f616e2d4370814a791" title="Click to view the MathML source">Cφclass="mathContainer hidden">class="mathCode"> to the Hardy–Smirnov space class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123615004577&_mathId=si7.gif&_user=111111111&_pii=S0022123615004577&_rdoc=1&_issn=00221236&md5=9f126bf538484c003b96ff9dff7f980b" title="Click to view the MathML source">E2(D)class="mathContainer hidden">class="mathCode"> in some domain class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123615004577&_mathId=si8.gif&_user=111111111&_pii=S0022123615004577&_rdoc=1&_issn=00221236&md5=d5248f8f42091317f6b25a06b5deba14" title="Click to view the MathML source">D⊃Dclass="mathContainer hidden">class="mathCode">. In particular, we obtain a characterization of Schatten membership of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123615004577&_mathId=si6.gif&_user=111111111&_pii=S0022123615004577&_rdoc=1&_issn=00221236&md5=6dc8dc4c25f403f616e2d4370814a791" title="Click to view the MathML source">Cφclass="mathContainer hidden">class="mathCode"> in terms of Nevanlinna counting function. By example this characterization does not hold true for general ϑ.