diff --git a/src/core/PocomathInstance.mjs b/src/core/PocomathInstance.mjs
index 1fce6b3..0253de0 100644
--- a/src/core/PocomathInstance.mjs
+++ b/src/core/PocomathInstance.mjs
@@ -7,6 +7,7 @@ import {typesOfSignature, subsetOfKeys} from './utils.mjs'
 const anySpec = {} // fixed dummy specification of 'any' type
 
 const theTemplateParam = 'T' // First pass: only allow this one exact parameter
+const templateFromParam = 'U' // For defining covariant conversions
 
 /* Returns a new signature just like sig but with the parameter replaced by
  * the type
@@ -803,11 +804,91 @@ export default class PocomathInstance {
       imps[signature] = does(refs)
    }
 
-   /* HERE!! This function needs to analyze the template and make sure the
+   /* This function analyzes the template and makes sure the
     * instantiations of it for type and all prior types of type are present
-    * in the instance
+    * in the instance.
     */
    _ensureTemplateTypes(template, type) {
+      let [base, arg] = template.split('<', 2)
+      arg = arg.slice(0,-1)
+      if (!arg) {
+         throw new Error(
+            'Implementation error in _ensureTemplateTypes', template, type)
+      }
+      let instantiations
+      if (this._templateParam(arg)) { // 1st-level template
+         instantiations = new Set(this._priorTypes(type))
+         instantiations.add(type)
+      } else { // nested template
+         instantiations = this._ensureTemplateTypes(arg, type)
+      }
+      const resultingTypes = new Set()
+      for (const iType of instantiations) {
+         const resultType = this._maybeAddTemplateType(base, iType)
+         if (resultType) resultingTypes.push(resultType)
+      }
+      return resultingTypes
+   }
+
+   /* Maybe add the instantiation of template type base with argument tyoe
+    * instantiator to the Types of this instance, if it hasn't happened already.
+    * Returns the name of the type if added, false otherwise.
+    */
+   _maybeAddTemplateType(base, instantiator) {
+      const wantsType = `${base}<${instantiator}>`
+      if (wantsType in this.Types) return false
+      // OK, need to generate the type from the template
+      // Set up refines, before, test, and from
+      const newTypeSpec = {}
+      const template = this._templateTypes[base]
+      if (!template) {
+         throw new Error(
+            `Implementor error in _maybeAddTemplateType ${base} ${instantiator}`)
+      }
+      const instantiatorSpec = this.Types[instantiator]
+      if (instantiatorSpec.refines) {
+         // Assuming all templates are covariant, for now
+         newTypeSpec.refines = `${base}<${instantiatorSpec.refines}>`
+      }
+      let beforeTypes = []
+      if (instantiatorSpec.before) {
+         beforeTypes = instantiatorSpec.before.map(type => `${base}<${type}>`)
+      }
+      if (template.before) {
+         for (const beforeTmpl of template.before) {
+            beforeTypes.push(
+               substituteInSig(beforeTmpl, theTemplateParam, instantiator))
+         }
+      }
+      if (beforeTypes.length > 0) {
+         newTypeSpec.before = beforeTypes
+      }
+      newTypeSpec.test = template.test(instantiatorSpec.test)
+      if (template.from) {
+         newTypeSpec.from = {}
+         for (let source in template.from) {
+            source = substituteInSig(source, theTemplateParam, instantiator)
+            const usesFromParam = false
+            for (const word of source.split(/[<>]/)) {
+               if (word === templateFromParam) {
+                  usesFromParam = true
+                  break
+               }
+            }
+            if (usesFromParam) {
+               for (const iFrom in instantiatorSpec.from) {
+                  const finalSource = substituteInSig(
+                     source, templateFromParam, iFrom)
+                  newTypeSpec[finalSource] = template.from[source](
+                     instantiatorSpec.from[iFrom])
+               }
+            } else {
+               newTypeSpec[source] = template.from[source]
+            }
+         }
+      }
+      this.installType(wantsType, newTypeSpec)
+      return wantsType
    }
 
 }